New characterization of Robertson-Walker geometries involving a single timelike curve

Our aim in this paper is two-fold. We establish a novel geometric characterization of the Roberson-Walker (RW) spacetime and, along the process, we find a canonical form of the RW metric associated to an arbitrary timelike curve and an arbitrary space frame. A known characterization establishes that a spacetime foliated by constant curvature leaves whose orthogonal flow (the cosmological flow) is geodesic, shear-free, and with constant expansion on each leaf, is RW. We generalize this characterization by relaxing the condition on the expansion. We show it suffices to demand that the spatial gradient and Laplacian of the expansion on a single arbitrary timelike curve vanish. In General Relativity these local conditions are equivalent to demanding that the energy flux measured by the cosmological flow, as well as its divergence, are zero on a single arbitrary timelike curve. The proof allows us to construct canonically adapted coordinates to the arbitrary curve, thus well-fitted to an observer with an arbitrary motion with respect to the cosmological flow.


Introduction
The basis for the standard model of the Universe is the Robertson-Walker (RW) spacetime.
The geometric nature of gravity makes the geometric characterization of spacetimes of great importance.Many efforts have been put in obtaining physically meaningful characterizations of the RW spacetime.Different characterizations probe different aspects the theory.Thus, finding new characterizations may be valuable to examine the foundations of the model.We refer the reader to the extensive review in [1].
The characterizations of the RW geometry involve several kinds of ingredients with different geometrical/physical/observational weight.The first basic ingredient is the existence of isometries [2,3].Another key ingredient is "kinematical", in terms of the existence of a family of fundamental observers defined as the (cosmological) flow U moving with the average matter in the Universe, to which certain requirements are imposed.The dynamical ingredients are represented by the Einstein field equations once the energy-momentum tensor T is constrained by any means.Other ingredients, that are ultimately linked with the above, involve conditions on the Weyl tensor or on the relation between the angular-diameter (or luminosity) distance and redshift [4].The latter is an example where observational constraints due to the isotropy of the measurements can be used to restrict the model [5].Another example is the isotropy of the Cosmic Microwave Background radiation that leads to the Ehlers-Geren-Sachs [6] characterization of RW.
The different known characterizations resort on a combination of various of the above items depending on which observational facts and principles, e.g.Cosmological or Copernican, are used to motivate them.The paradigmatic characterization is due to Ellis [7] and states that RW is characterized by the energy-momentum T being isotropic (and thus of a perfect fluid type) and the flow U along the timelike eigendirection of T being irrotational, shear-free and geodesic.Another characterization (see [8] [Section 1.3.2])that will play an important role later states that a spacetime that (a) is foliated by spaces of constant curvature and (b) the orthogonal flow is geodesic, shear-free and the expansion is constant on each leaf, must be RW.
Let us note that in some cases not all the assumptions involved in the characterizations are spelled out in full detail or with sufficient accuracy.Regarding the use of isometries, we refer the reader to [9] for a critical analysis on the characterizations based mainly on the existence of different notions of isotropies.
It must be also stressed that, although the aim of the mentioned results (and the present work) is to characterize FLRW, there exist alternatives to the Cartan-Karlhede characterization scheme (see Chapter 9 in [3]) or modifications thereof that provide, in particular, characterizations for different families of FLRW geometries.That can be used to consider the equivalence problem between different FLRW.The IDEAL1 characterization of FLRW [11] is based entirely on Rainich-type tensorial equations (see [12] in a more general setting) without relying on the geometric structure of any (timelike) congruence.On the other hand, the classification algorithm constructed in [13] is based on invariant quantities relative to a congruence of fundamental observers.
All characterizations based on isotropy properties around one single fundamental observer rely ultimately on the Copernican principle in order to extend these properties to all fundamental observers, thus to the cosmological flow U.In this way, the properties are made global throughout the Universe and the conclusion that the geometry is RW everywhere can be reached.In fact, all characterizations of RW spaces need a certain amount of global properties.In this paper we aim at finding a characterization result that splits the hypotheses in two sets, namely conditions that hold everywhere and conditions that hold on a single curve.As already said, global hypotheses are unavoidable, but in this paper we aim at weakening them.The assumption of global type that we make concern the geometry of the slices of constant cosmological time, i.e. the spaces orthogonal to U. Specifically we require that every slice at constant time t has constant curvature k(t), that its second fundamental form is pure trace and that the cosmological time has constant lapse.The last two conditions can be stated equivalently in terms of U by demanding that this field is shear-free and geodesic (compare with item (b) in the characterization above).Let us stress that the curvature of the slices as a function of time k(t) is left free, and can, in principle, change sign.This fact is clearly consistent locally, and there are even explicit global smooth constructions [14].
The distinctive feature of our characterization is the presence of a local condition based on a single timelike curve o.This curve is completely arbitrary so it need not correspond to a fundamental observer (although, of course, it may be).In fact, we can even dispense of the condition that the curve is timelike and consider any curve with the only restriction that it is always transverse to the cosmological time.The condition that we impose is that the mean curvature of the constant time slices have vanishing gradient and vanishing Laplacian along o.An equivalent way of stating this is that the expansion of the cosmological flow U has vanishing gradient and vanishing Laplacian along the chosen curve.Note that this condition is much weaker than imposing that the expansion is homogeneous in space.Interestingly, the local condition that we impose on the curve o can also be stated as a restriction on the energymomentum contents.Assuming the validity of the Einstein field equations, this condition can be rephrased as demanding that the energy flux measured by the fundamental observer along the curve o vanishes, together with its divergence.We find it remarkable that the global assumptions can be relaxed significantly and the extra parts be replaced by a very weak and physically reasonable condition on a single curve.
The precise statement of this characterization result appears in Theorem 4.2 below.We provide here a slightly more informal version.
Theorem 1.1.Let (M, g) be a spacetime of the form M = I × Σ, where I ⊂ R is an open interval.Assume that each Σ t := {t} × Σ is spacelike and let g t be the induced metric and U its future directed unit normal.Assume that g t has constant curvature k(t) and that U is shear-free and geodesic.Then (M, g) is a Robertson-Walker space if and only if there is a timelike curve o(t) along which the expansion Θ of U satisfies The main idea behind the proof of this theorem is to show that the local condition along the curve together with the global assumptions are sufficient to prove that that expansion Θ of U must be constant on each slice Σ t .We can then rely on the known characterization mentioned above to reach the conclusion.
The method of proof is based on the construction of stereographic coordinates at each Σ t centered at the point o(t) and associated uniquely to an arbitrary choice of orthonormal frame {X A (t)} at o(t).These coordinates not only play a crucial role in the proof, but they also have an added bonus.Once the characterization result is obtained, we have as a byproduct the form of the RW metric written in terms of coordinates canonically constructed from an arbitrary timelike curve o(t) (actually any tranverse curve) and an arbitrary choice of orthonormal space frame {X A (t)} along o.The form of the metric involves, besides the scale factor a(t) and the discrete curvature parameter ǫ 0 ∈ {−1, 0, 1}, three2 functions F 0A (t) and three functions F AB (t) = −F BA (t) associated to the freedom in the curve o(t) and the frame X A (t).In fact, the functions F 0A describe the velocity of the observer along the curve with respect to the cosmological flow U, and we show that by an appropriate choice of frame we can always set F AB (t) = 0 if so desired.The precise form of the metric in these coordinates is the content of Theorem 4.5.The form of the metric is of course more complicated than any of the standard ones, but the fact that it is canonically adapted to a single arbitrary observer makes it interesting and potentially useful in problems in Cosmology where a privileged observer not at rest with the cosmological flow is involved, as it happens with our direct observations [15] (e.g.any Earth-based or satellite telescope).To be precise, the dipole contribution to the CMB due to our peculiar velocity with respect to the cosmological flow can thus be incorporated at the background level of a perturbative approach by setting the peculiar velocity as the three- the coordinates of Theorem 4.5 (see Remark 4.7).That could help on the disentangling the degeneracy of the Doppler effect, due to the peculiar velocity and a dipolar part of the perturbations (see e.g.[16]).
The plan of the paper is as follows.In Section 2 we give the precise definition of RW space that we shall use.The definition is standard except that we allow for both Lorentzian and Riemannian signature, since this generality entails essentially no extra effort.We then quote a known characterization result of RW geometries that will play a key role in our argument.In Section 3 we describe the geometry of spaces that admit a foliation of constant curvature and umbilic leaves.The geometric framework in this section is more general than in the rest of the paper.Although we could have simplified this part, there are several applications that we have in mind where the more general framework is needed.The key result of this section is Proposition 3.3 where canonical coordinates adapted to a transverse curve o(t) and an orthonormal frame are obtained.On each leaf of the foliation these coordinates are stereographic, so in order to make the paper self-contained we provide in Appendix A a very simple and direct construction of stereographic coordinates on any Riemannian space of constant curvature.Section 4 contains our main results.Theorem 4.2 is the characterization result of RW involving a transverse curve.Theorem 4.5 provides the RW metric in canonical coordinates adapted to {o(t)} and {X A (t)}.Subsection 4.1 is devoted to describing several geometric properties of these coordinates, in particular the relation of the functions F AB (t) to rotations of the frame {X A (t)}.In Subsection 4.2 the explicit form the Killing vectors of RW in the newly constructed coordinates is presented.

Basic notation
The set of vector fields on a manifold U is denoted by X(U).We use £ X for the Lie derivative along a vector field X ∈ X(U).Given a function f ∈ C ∞ (M, R) in a semi-Riemannian space (M, g) we use grad g f to denote its gradient vector field and ∆ g f for the Laplacian.We use both abstract-index or index-free notation depending on our convenience.Capital Latin indices A, B, C take values 1, • • • , n and Greek indices in 0, • • • , n + 1.The symbol δ AB stands for the Kronecker delta.

Definitions and a previous characterization result
In this paper we adopt the following definition of an n + 1 dimensional RW geometry: Definition 2.1 (RW space).Let ε = ±1, I ⊂ R an open interval and (Σ, g ǫ 0 ) a (positive definite) Riemannian manifold of dimension n ≥ 1 and constant curvature ǫ 0 ∈ {−1, 0, 1} 3 .A RW space, denoted by εI × a Σ, is the manifold M = I × Σ endowed with the warped product metric g = −εdτ 2 + (a • τ ) 2 g ǫ 0 , where τ ∈ C ∞ (M, I) is the projection to the first factor and a ∈ C ∞ (I, R + ) is the warping function.
This definition is standard except for the fact that we are allowing any sign in ε, which means that g can be either of Lorentzian (ε = 1) or Riemannian (ε = −1) signature.Our primary interest is in the Lorentzian case, but with essentially the same effort we can deal with both cases.Note that we are not imposing any global assumption on the base space (Σ, g ǫ 0 ), such as completeness or simply connectednes.
This definition is sufficiently general for our characterization purposes, and it is also welladapted to a known characterization result that will play a relevant role later.Characterization results can be either local or global.One speaks of a local characterization when the assumptions made on the space (M, g) under consideration are sufficient to show that there exists a RW space (M RW , g RW ) such that, at every point p ∈ M there is an open neighbourhood of W p of p and a map Φ p : (W p , g| Wp ) → (M RW , g RW ) which is an isometry from (W p , g| Wp ) onto its image.One simply says that (M, g) is locally a RW space.The characterization is global if there exists an isometry Φ : (M, g) → (M RW , g RW ).Obviously, global characterizations are stronger, so they also need stronger assumptions.
There are several local and global characterization results of RW spaces.We refer to [9] for a detailed recent account that discusses many of them (and corrects some misleading or incorrect statements in the literature).The one that will be relevant for this paper was stated in rigorous form in [17] in the more general context of generalized RW spaces.This result had a local and a global version.The global version was then improved in [18] building on previous results in [19].The statements of those papers needed here are summarized in the following theorem (see also Proposition 1 in [20], and [21], [8]).
Suppose that the hypersurfaces defined by the level sets of τ are of constant curvature.Then (M, g) is locally a RW space.Assume further that, for some t 0 ∈ R there exists an interval I ⊂ R such that the map Φ : I × Σ t 0 → M defined as the flow of the vector field U is well-defined and onto.Then (M, g) is globally a RW space. 4emark 2.3.Note that, because of (i) and (ii), the covector U := g(U, •) satisfies dU = 0, and therefore there is a function τ so that U = grad g τ , locally.
Remark 2.4.Theorem 3.1 in [18] is stated differently.Hypotheses up to (iii) are replaced by the condition that (M, g) admits a gradient timelike conformal Killing vector field, namely a vector field U ′ which is the gradient of a function τ ′ ∈ C ∞ (M, R) (in general, different from τ above) and satisfies It is easy to show that the two sets of conditions are equivalent.
3 Umbilic foliations with constant-curvature leaves Although our characterization theorem involves Lorentzian or Riemannian spaces, it is of interest to derive some of the intermediate results in a more general setup.As already advanced in the Introduction, this is convenient for future applications to other problems.In this section we describe this more general setup.Throughout this section (U, I, τ ) denotes an (n + 1)-dimensional manifold, I ⊂ R an open interval and τ a smooth function τ : U → I such that dτ does not vanish anywhere.For each t ∈ I the level surface of τ , Σ t := τ −1 (t) is a hypersurface of U and the set of all {Σ t } foliate U (see e.g.[2]).The submanifolds Σ t are always embedded, but in general they not need to be diffeomorphic to each other, or connected.However, in sufficiently small open neighbourhoods of a point p ∈ U this is always the case.We call (U, I, τ ) a foliated space.
The following notation will be used in foliated spaces.The symbol t always denotes a value in I.For any f ∈ C ∞ (U, R), we use f t for the restriction of f to Σ t .More precisely f t := i ⋆ t (f ) where i t : Σ t ֒→ U is the inclusion map.Any function F ∈ C ∞ (I, R) can be transferred to a function on U which we denote with the same symbol but in boldface font.Specifically, F := F •τ .A vector field X ∈ X(U) is said to be tangent to the foliation iff X(τ ) = 0.For such vector fields, there always exists a unique vector field We shall consider foliated spaces (U, I, τ ) that carry an additional geometric structure capable of inducing a positive definite Riemannian metric on each Σ t .We therefore assume that U is endowed with a symmetric two-covariant tensor field Υ such that Υ t := i ⋆ t (Υ) is a positive definite metric.For this paper, it would suffice to assume that Υ is either a Riemannian metric, or a Lorentzian metric for which Σ t are spacelike hypersurfaces, but we keep this more general setup for future purposes.We call the collection (U, I, τ, Υ) a metric foliated space.
In a metric foliated space there is a unique vector field V defined by the properties (i) V (τ ) = 1 and (ii) Υ(V, X) = 0 for all vector fields X ∈ X(U) tangent to the foliation.Indeed, at any point p ∈ U the tensor Υ| p is positive definite when restricted to the n-dimensional subspace of vectors tangent to Σ t .Thus, the signature of Υ| p is necessarily {ǫ p , +, • • • , +} where ǫ p ∈ {−1, 0, 1} (note that ǫ p may change from point to point).By basic linear algebra, the space is one-dimensional.All non-zero vectors Z ∈ (T p Σ t ) ⊥ are transverse to Σ t , i.e. satisfy Z(τ ) = 0 (otherwise we would have a non-zero vector tangent to Σ t and orthogonal to all other tangent vectors, which is impossible given that Υ t is positive definite).It is therefore clear that there is exactly one vector It is immediate to check that V depends smoothly on p and hence defines a vector field.We shall call V the normal vector field of the foliation.Observe that V depends on the choice of τ and not just on the leaves {Σ t } of the foliation.Another function τ ′ : U → I ′ will define exactly the same leaves as τ if and only if τ ′ = F •τ where F : . This property will be used below.To avoid any misunderstanding, note that we are making no assumption on the norm of V .Even when Υ is a metric, V will in general not be a unit normal to the hypersurfaces.
The two global assumptions we shall make in order to characterize a RW space are encoded in the following definition: Definition 3.1.A metric foliated space (U, τ, I, Υ) is called an umbilic foliation with constant curvature leaves iff: 1.There exist a smooth function κ : for all vectors X, Y tangent to the foliation.
2. The induced metric Υ t on each Σ t is a metric of constant curvature k(t).
It will be convenient to distinguish when conditions 1 or 2 are being used.When only 1 holds we speak of an umbilic foliation and when only 2 holds of a foliation with constant curvature leaves.
As described in the Introduction, the local assumption that we make to characterize a RW geometry involves a single timelike curve.In the more general setup of this section we consider a smooth curve o transverse to the foliation, i.e. such that its tangent vector t o satisfies t o (τ ) = 0 everywhere.Letting I o ⊂ I be the image of τ • o (i.e. the set of values that τ takes along the curve), we may parametrize o by t ∈ I o .In other words, we consider the curve as described by a map o : By definition we say that the curve o is parametrized by t whenever this happens.
The curve o describes the path followed by our privileged observer.We also need this observer to be endowed with a frame.To that aim, we select n vector fields From the curve o and the orthonormal frame {X A (t)} we can construct a canonical coordinate system in a neighbourhood of o as follows.For each t ∈ I o we have a point o(t) ∈ Σ t and an orthonormal frame {X A (t)} of T o(t) Σ t .The space (Σ t , Υ t ) is a space of constant curvature k(t).In appendix A we give a self-contained and direct description of how to construct stereographic coordinates of (Σ t , Υ t ) centered at o(t) and with frame {X A (t)}.The procedure involves a so-called Hessian basis (see Definition A.1), which is a set of n + 2 real valued functions i.e. we simply stack the functions {Y α t } together.Since the boundary conditions are independent of t (in particular smooth in t) and the curve o is also smooth, it follows that the functions {Y α } are smooth on U ′ .
For each value t ∈ I o , we now apply Lemma A.2 (together with Remark A.5) to (S, As a consequence, the functions {λ A } defined by are well-defined on some open domain V o ⊂ U ′ containing o.We restrict ourselves to V o from now on (and still use Σ t to denote Σ t ∩ V o ).
In terms of λ A the functions {Y α } take the form, c.f. (A.15), where |λ| 2 := δ AB λ A λ B since in the present case h AB = δ AB given that the basis {X A } has been taken to be orthonormal.The restriction λ A t of λ A to each Σ t are stereographic coordinates of Σ t with center o(t) and frame {X A (t)}, so the metric takes the form (cf. (A.8)) The functions {Y α } are smooth on V o , so the same holds for {λ A }.Moreover, the set {τ, λ A } defines a coordinate chart on V o .Since we know the form of the restriction of Υ to the hypersurface {τ = t} ∩ V o , namely (3.5), the full tensor Υ in coordinates {τ, λ A } must take the form We can also express the normal vector field of the foliation V in this coordinate chart.The two conditions V (τ ) = 1 and Υ(V, Hence f A = −V (λ A ) and we can rewrite (3.6) as This expression gives us a handle on how to determine the form of the tensor Υ in the context of Definition 3.1.So far we have only assumed that the foliation has constant curvature leaves.We now impose the condition that the foliation is umbilic.The stereographic coordinates have allowed us to construct a vector field ∂ τ on V o .This vector field is canonical in the following sense.In geometric terms ∂ τ is the field of tangents to the curves of constant λ A parametrized by τ .Since the stereographic coordinates are canonically constructed from the curve o and the basis {X A (t)}, the vector field depends only on o and {X A (t)}.It can therefore be denoted by W o,X .We prefer this name over ∂ τ , as this emphasizes its geometric meaning.Obviously, in the coordinates {τ, λ A } it holds W o,X = ∂ τ .The following proposition is key in the determination of Υ. Proposition 3.2.Let (U, I, τ, Υ) be an umbilic foliation with constant curvature leaves as defined in Definition 3.1.Let o be a curve parametrized by t and {X A (t)} an orthonormal basis along o.Consider the domain V o as constructed above.
Then the vector field V − W o,X defined on V o is tangent to the foliation and the restriction to each leaf Σ t is a conformal Killing vector of Υ t with conformal factor κ t + Y n+1 t k(t) (c.f.(3.8)).
Proof.The fact that V − W o,X is tangent to the foliation is immediate from (V − W o,X )(τ ) = 0. Now we compute £ W o,X Υ acting on tangent vectors.In the coordinates {τ, λ A } we have that W o,X = ∂ τ and that ∂ A is a basis of tangent vectors.So, it suffices to compute where [∂ τ , ∂ λ A ] = 0 was used in the first equality and (3.7) in the second.In more geometric terms, this equality can be written as Concerning V , the umbilicity condition (3.1) can be written as Now, for any vector field Z tangent to the foliation, i.e. of the form Z = di t (Z t ), Z t ∈ X(Σ t ), and any covariant tensor field T on U the following general identity holds Applying this to Z = V − W o,X and T = Υ, and defining which proves that V = t is a conformal Killing vector of Υ t with conformal factor κ t + Y n+1 t k(t), as claimed.
In Appendix A we write down a basis of the conformal Killing algebra of any space of constant curvature in terms of a Hessian basis (related results can be found in [22]) and we find the explicit form of the basis vectors in stereographic coordinates centered at a point o with basis {X A }.This requires that the dimension of the space is at least three.So we assume n ≥ 3 from now on.
Translating the results in A.
Moreover, these fields satisfy so in particular {η 0A t , η AB t } with A < B is a basis of the Killing algebra of (Σ t , Υ t ).Let us define the vector fields η αβ on V o by di t (η αβ t ).In the coordinates {τ, λ A } we clearly have 12) The conformal Killing vector V = t must satisfy (3.8), which after using (3.9) on the left hand side, yields This restriction links the umbilicity function κ and the rate of change of the space curvature to the functions {C 0 , C A }.The other functions {F 0A , F AB } in (3.12) are associated with the isometries of each leaf, and they remain completely free.
The geometrical meaning of the functions {C 0 , C A } can be determined by evaluating (3.13) as well as its derivatives along X A (t), along the curve o.Since in the coordinates {τ, λ A } this curve is simply {τ = t, λ A = 0} and the orthonormal basis is Proof.Another way of writing equation (3.13) is which becomes (3.16) after we insert (3.4).To show (3.17) we first note that (3.12) gives ) after recalling that W o,X = ∂ τ in the coordinates {∂ τ , ∂ A } and inserting the explicit expressions (3.10)- (3.11).Using this in (3.7) brings Υ into the form (3.17).Since the stereographic coordinates {λ A } are unique once the curve o and the orthonormal frame {X A } along o has been chosen, the various uniqueness claims hold.
Remark 3.4.The neighbourhood V o extends as far as the stereographic coordinates λ A around the curve o can be defined.This property will be used below.
So far we have found necessary consequences of Definition 3.1.To make sure that no other information can be extracted we need a reciprocal result.Proposition 3.5.Let (V o , I o , τ, Υ) be defined by V o := I o × B 0 (r), where B 0 (r) is a ball in R n centered at 0 with sufficiently small radius r, τ ∈ C ∞ (V o , R) is the projection of I o × B 0 (r) onto the first factor, and h is given by (3.17) where {λ A } are Cartesian coordinates on B 0 (r).Then (V o , I o , τ, Υ) is an umbilic foliation with constant curvature leaves with umbilicity function given by (3.16) and curvature k(t) := k t .
Proof.It is immediate to check that (V o , I o , τ, Υ) is a metric foliation.Condition 2 in Definition 3.1 is also immediate.For condition 1 note that the normal vector of the foliation is given by (3.18), hence also (3.12) holds and the validity of (3.1) with κ given in (3.16) follows easily from the fact that η αβ t are conformal Killing vectors satisfying (3.9).

RW foliation
For our characterization purposes, it is of interest to find sufficient conditions along the curve o that guarantee that the umbilicity function κ is homogeneous, i.e. constant on each leaf Σ t .To that aim, observe that the umbilicity function (3.16) is the solution of the equation (3.15) with boundary data (3.14).It is obvious from the explicit form in (3.16) that κ is homogeneous if and only if C A = 0 and 2C 0 k + k = 0. We translate this into two geometric conditions along the curve o.We shall say that Condition RW.1 (resp.RW.The following result determines the form of Υ in the case when the geometric conditions RW.1 and RW.2 hold.Proposition 3.7 (RW foliation).Assume the setup of Proposition 3.3 and, in addition, that the geometric conditions RW.1 and RW.2 hold.Then, there exist coordinates {τ, ) where |z| 2 := δ AB z A z B .In addition, the curve o is given by {τ = t, z A = 0} and the frame Proof.By Lemma 3.6 we have C A = 0, C 0 = 1 2 Ξ, and (3.19) holds.Inserting into (3.17)yields ).Thus k = ǫ 0 a −2 for some constant ǫ 0 .Since After defining ω A := a −1 θ A , the form of the metric (3.21) follows.Clearly, the curve o(t) is defined by {τ = t, z A = 0} and the claim on the basis {X A (t)} follows from Observe that the form of the normal vector of the foliation V in the coordinates {τ, z A } can be obtained directly from (3.20) and reads Note that ∂ τ in this expression is not the geometrically defined vector W o,X because ∂ τ is a different vector field in the coordinates {τ, λ A } than in the coordinates {τ, z A }. Explicitly, in the latter coordinates we have and An umbilic foliation (V o , I o , τ, Υ) for which Proposition 3.7 holds will be called a RW foliation.4 New characterization of RW geometries In the previous section the tensor Υ was allowed to be degenerate, Lorentzian or Riemannian depending on the point.This is why we could say nothing about the function Q that appears in (3.17).Indeed, this function is related to V by means of (this follows immediately from θ A (V ) = 0, cf.(3.17) and (3.18), and dτ (V ) = 1).Thus, in order to restrict Q we need to make further assumptions on Υ.
In this section we assume that Υ is a metric.To emphasize this fact we replace Υ by g from now on.Although our main interest is when g is Lorentzian we still allow for the Riemannian case as this entails no extra effort.g being a metric is equivalent to g(V, V ) being nowhere zero.Thus, there exists a positive function H ∈ C ∞ (U, R) and a sign ε ∈ {−1, 1} such that The space (U, g) is a semi-Riemannian manifold with signature {−ε, 1, • • • , 1}.Since Σ t are now spacelike hypersurfaces of (U, g), they admit a unique unit normal U pointing along increasing values of τ , i.e. satisfying g(U, U) = −ε, U(τ ) > 0.
By construction V and U are proportional to each other (because both are g-orthogonal to T p Σ τ (p) for all p ∈ U).Combining with (4.2) it follows that For any vector X we denote by X the covector obtained by lowering indices with g.It follows directly from its definition that V is proportional to dτ and, in fact, because writing V = P dτ we have Consequently, U = −εHdτ. (4.5) The hypersurface Σ t admits a second fundamental form K t which can be computed using Thus, where we applied the straightforward identity together with (4.3).By (4.7), the notion of (U, I, τ, g) being an umbilic foliation, cf.(3.1), is equivalent to the second fundamental form K t of Σ t being pure trace (also called totally umbilic or shear-free).In such case, the second fundamental form reads Note that the class of spacetimes given by (U, g) produces a class of cosmological models with spatial constant curvature k(t) that can change sign from leaf to leaf.Observe that in a RW cosmology the sign of k(t) cannot change.The class (U, g) thus belongs to the "SI" (spaceisotropic) type in the classification produced in [9], where the original family of spacetimes recently constructed in [14] is analysed as a paradigmatic example.Now, recall the expansion (also called mean curvature) of a hypersurface is the trace of its second fundamental form.It is of interest to find the most general form of g when not only K t is pure trace, but in addition the expansion is homogeneous and non-zero, i.e. of the form θ for some smooth nowhere zero function θ ∈ C ∞ (I, R).Proposition 4.1 (Homogeneous expansion).Let (U, g) be a semi-Riemannian manifold of dimension n+1, n ≥ 3, and signature {−ε, +1, • • • , +1} endowed with a function τ ∈ C ∞ (U, I), for some interval I ⊂ R, satisfying dτ = 0 everywhere.Let Σ t be the level sets of τ and assume that: (i) The induced metric g t on each Σ t is of constant curvature k(t).
(ii) There is a nowhere zero function θ ∈ C ∞ (I, R) such that the second fundamental form K t of Σ t is of the form Select any curve o parametrized by t and an orthonormal frame {X A (t)} of T o(t) Σ t smoothly depending on t.Then, there is a neighbourhood V o of o and unique coordinates {τ, λ A } on V o such that the metric g takes the form (3.17) with Q given by Since Q = −εH 2 and κ is given by (3.16) the result follows.
We are ready to state and prove our main results of the paper.The only extra restriction we need is that the unit normal vector U is geodesic.In the first theorem we find necessary and sufficient conditions for any metric admitting a foliation by totally umbilic hypersurfaces of constant curvature and with geodesic unit normal to be locally isometric to a RW space.Again we allow for both Lorentzian and Riemannian signatures.In the second theorem we write down the RW metric in canonical coordinates adapted to an arbitrary transverse curve carrying an orthonormal frame adapted to the foliation.Theorem 4.2 (Local characterization of RW in terms of a curve).Let (U, g) be a semi-Riemannian manifold of dimension n + 1, n ≥ 3, and signature {−ε, +1, • • • , +1} endowed with a function τ ∈ C ∞ (U, I), for some interval I ⊂ R, satisfying dτ = 0 everywhere.Let Σ t be the level sets of τ and assume that each Σ t is connected and that: (i) The induced metric g t on each Σ t is of constant curvature k(t).
(ii) The second fundamental form of Σ t is pure trace, i.e. there is a function Then, (U, g) is locally a RW space if and only if there exists a curve o parametrized by t and defined all over I such that the expansion Θ along the curve satisfies Proof.We start with the "only if" part, i.e. we assume that (U, g) is locally isometric to RW.The expansion Θ in a RW space is constant on each orbit of the isometry group, so for any transversal curve o conditions (4.12) hold.
For the "if" part, we denote by ∇ g i the Levi-Civita derivative of g and use abstract indices i, j, . . . on U. By items (i) and (ii), (U, I, τ, g) is an umbilic foliation with constant curvature leaves (cf.Definition 3.1) and the unit normal U to the leaves Σ t is given by (4.5).We compute the acceleration of U as follows (see e.g.[23]) We can now take any other curve o ′ parametrized by t, and defined all over I or any subinterval thereof, in the domain V o .By the "only if" part of the theorem, conditions (4.12) hold also for o ′ , and we can thus construct its corresponding domain V ′ o .By homogeneity of the leaves, the construction of the stereographic coordinates {z A } in Proposition 3.7 (see Lemma A.2) is independent of the curve.This implies, in particular, that for any t ∈ I there exists a sufficiently small r(t) > 0 such that the geodesic ball B p (r(t)) of radius r(t) centered at p ∈ Σ t is covered by the stereographic coordinates λ A with origin at p. Note that r(t) is independent of p ∈ Σ t .By Remark 3.4 the neighbourhood V ′ o contains B o ′ (t) (r(t)).Since each Σ t is connected, it is now clear that we can reach any point by overlapping domains V o for different curves, and therefore cover the whole of U. As a result the unit normal U to Σ t satisfies all conditions (i), (ii), (iii) of Theorem 2.2 and the result follows.
The conclusion of the theorem, namely that (U, g) is locally a RW space, cannot be strengthed because our global 6 assumptions on (U, g) are too weak to conclude more.Indeed, one can remove any open set from (U, g) away from the curve o and all the hypothesis of the theorem still hold.This prevents (U, g) from being globally isometric to a RW space.To get a global result we need additional hypotheses.We write down one such result based on the global characterization in [18], quoted in Theorem 2.2 above.Theorem 4.3 (Global characterization of RW in terms of a curve).Let (U, g) satisfy the hypotheses of Theorem 4.2 and let U be the vector field defined in its item (iii).Assume that for some t 0 ∈ R there exists an interval I ⊂ R such that the map Φ : I × Σ t 0 → M defined to be the flow of the vector field U is well-defined and onto.Then (U, g) is a RW space if and only if there exists a curve o : I → U parametrized by t such that the expansion Θ satisfies (4.12) for all t ∈ I.
Proof.In the proof of Theorem 4.2 we have shown that (U, g) satisfies conditions (i) to (iii) of Theorem 2.2.Thus, the result follows from the global part of Theorem 2.2.
Remark 4.4.The purely local conditions that we are imposing along the curve o involve the expansion Θ of U, cf.equations (4.12).It is of interest to translate these conditions in terms of the energy-momentum contents.So, let us assume that we are in the context of Theorem 4.2 so that (U, g) satisfies conditions (i) to (iii) and, only for this remark, that the Einstein equations hold.The momentum constraint on the hypersurface Σ t is div where div gt is the divergence in (Σ t , g t ) and J t is the energy-flux with respect to the observer U at Σ t .Since the second fundamental form of Σ t is pure trace (cf.(4.11)), this equation becomes Thus, conditions (4.12) can be equivalently stated in terms of the energy flux by imposing that both J t and its divergence div gt J t vanish along the curve o.
An interesting by-product of the characterization of RW spaces is the construction of an explicit coordinate system canonically adapted to any transverse curve and to any orthonormal space-frame defined along the curve.This frame spans, at every instant of cosmological time, the corresponding cosmological rest space.Note that (in the Lorentzian setting) the curve is allowed to be of any causal character, so in general there is no local rest space for the curve itself.
This coordinate system could be relevant to study cosmological effects felt by a single observer moving arbitrarily with respect to the cosmological flow.Obviously, for such case it is necessary to restrict the transverse curve to be timelike.Theorem 4.5 (Coordinates in RW defined along an arbitrary transverse curve).Consider a RW space (M = I × Σ, g) of dimension n + 1, n ≥ 3, with scale factor a : I → R and curvature ǫ 0 , i.e.
where g ǫ 0 is a metric of constant curvature ǫ 0 on Σ.Let Σ t be the hypersurfaces {τ = t}.Select any smooth curve o : I → M satisfying o(t) ∈ Σ t and an orthonormal frame Then there exist coordinates {τ, z A } and functions where |z| 2 := δ AB z A z B .In addition, the curve o is given by {τ = t, z A = 0} and the frame Proof.The set (M, I, τ, g) is clearly a RW foliation and the second fundamental form of the constant curvature leaves Σ t = {τ = t}, t ∈ I, satisfies Proposition 3.7 provides the form of g = h and ω A , leaving the function Q to be determined.The proof of this proposition also gives C A = 0, C 0 = ȧ/a and k = ǫ 0 a −2 .Combining (4.16) with (4.9) gives Θ t = n ȧ(t)/a(t).We can now apply expression (4.10) in Proposition 4.1 to conclude Q = −ε and the theorem is proved.
Remark 4.6.Since Q = −ǫ the foliation function τ is such that dτ is unit.Thus H = 1 as a consequence of (4.5) and hence V = U by (4.3).Consequently the flow vector U in the coordinates of the theorem takes the form (3.24) Remark 4.7.In the Lorentzian case ε = 1, the curve o(t) is timelike iff F 0A (t)F 0B (t)δ AB < 1, and then it describes the single observer that moves with three-velocity with respect to the cosmological flow U. On the other hand, as expected, the functions F AB relate to the rotation of the frame {X A (t)} along the curve with respect to the cosmological flow.We devote Subsection 4.2 below to prove these statements and provide the explicit relations, in particular.
Remark 4.8.Theorem 4.5 requires n ≥ 3 because the core of our argument relies on using a finite dimensional basis of conformal Killing vectors, and this requires that the spaces have at least dimension three.However, one checks that when n = 1, 2 the metric (4.14)-(4.15) is still locally isometric to a RW space.This can be done in several different ways.A simple one is to note that g admits n(n + 1)/2 linearly independent Killing vector fields tangent to spacelike hypersurfaces.The explicit form of these Killing vectors is obtained in Subsection 4.2 below.

Description of the new coordinates
In this subsection we present a description of the geometric meaning of the free smooth functions F 0A (t) and F AB (t).Before doing so, however, it is worth to comment on the method that we have followed to determine the RW metric in the new coordinates.
In Section 3 we have applied the construction in Appendix A to build, in the whole foliated manifold, stereographic coordinates on each constant curvature leaf centered on an arbitrary curve o(t) crossing the foliation.The curvature on each leaf κ(t) was allowed to change sign.Since the transformation that sends stereographic coordinates centered at one point to stereographic coordinates centered at another point in a space of constant curvarture κ is an isometry, one could think of approaching the problem with a direct coordinate change, namely a transformation that sends leaves to leaves and that, restricted to each leaf, defines an isometry (with parameters depending on t).However, this approach has some disadvantages.First, the isometries take different forms depending on whether κ is zero or non-zero.In the former case the isometries are just the standard Euclidean motions and in the second one they are a subset of the Möbius transformations [24], so both cases cannot be treated in one go.Working on a sufficient local patch, the two cases can be treated simultaneously by means of rotations and so-called "quasitranslations" (see e.g.[25]).However, such coordinates do not cover the whole space when κ is positive (this is related to the last comment in Remark A.3 in AppendixA).So these methods cannot be applied to cases when the curvature is allowed to change sign and one wants to cover the whole manifold (up to the antipodal point, obviously).Even restricting oneself to foliations with everywhere positive, zero or negative curvature, the direct coordinate transformation method turns out to be remarkably more complicated that the approach that we have followed here.Without knowing beforehand the final form of the tensor (3.20) (or of the metric (4.14) in the RW case) it would have been quite difficult to show that the transformation that maps leaves into leaves and sends the origin into a moving point o(τ ) takes the final simple form that we find.In any case, in Remark 4.10 below we provide the relation of the parametrization of the curve o(τ ) expressed in standard RW comoving coordinates with the coefficients F 0A and F AB in the simplest possible case (corresponding to a certain rotation of the coordinates being set to the identity).
We now proceed with the description of F 0A in terms of the curve o and the orthonormal frame {X A (t)} selected along the curve.First, observe that the character (timelike, spacelike or null) of the curve o is completely arbitrary and may even depend on the point (of course this only makes sense for ε = 1).Since the vector tangent to o is W o,X | o(t) , and in the coordinates {τ, z A } of Theorem 4.5 one has Therefore, in the case ε = 1 the curve o is timelike at points where A (F 0A (t)) 2 < 1.On the other hand, since , we have Thus, F 0A (t) provide a measure of the angle between the leaves Σ t and the curve o at time t, and thus, how the projection of the curve o on the leaves moves with t.Explicitly, the threevelocity of the observer following the curve o with respect to the cosmological flow, defined as the standard of the projection of ∂ τ orthogonal to U at o, is thus explicitly given by as stated in Remark 4.7.
Assume now that o is nowhere null.Recall that the Fermi-Walker derivative of a vector field X(t) along a curve γ(t) with tangent vector γ is defined as and that it describes the rate of rotation of the field X(t) along the curve.It is therefore of interest to compute D F W o X along the curve o.The computation is aided by the fact that the tangent vector of o is the restriction to o of the vector field W o,X .We start with the Fermi-Walter derivative of the flow vector U.A straightforward calculation yields where here and in the following we use δ AB to raise the indexes A, B, . ... The Fermi-Walker derivative of the frame vectors X A (t) is most easily expressed in terms of the functions Z(z A ), namely the A-components of the vector Z in the basis {∂ τ , ∂ z A }.The result is Observe that demanding that the frame {X A (t)} is transported Fermi-Walker along o restricts the curve itself to be at a constant angle F 0A (t).This may seem strange at first sight because one could think that the freedom in choosing the frame should allow for any type of Fermi-Walker transport along any curve.However, this is not the case because by construction the frame {X A (t)} has already been restricted to be tangent to the leaves, which prevents one from choosing an arbitrary Fermi-Walker transport law along o.
It is also of interest to study the consequences of imposing that the flow vector U be transported Fermi-Walker along o, i.e.Z = 0. Again, this restricts the form of o.Inserting Z = 0 in (4.18) gives d dt In any case, since the Fermi-Walker transport of a frame is associated to its rotation along the curve, we have thus clear indication that F AB is related to rotation.We confirm this by showing that we can get rid of F AB (t) by means of a rotation of the frame {X A (t)} along any curve o(t).
A rotation of the frame is a transformation of the form Let us first show how a rotation translates to a change of the corresponding stereographic coordinates.Since the coordinates z ′A are adapted to X ′ A (t), the transformation above is equivalent to . By uniqueness of the stereographic coordinates, see Remark A.4, the change of coordinates must satisfy and, given that o is located at z ′A = 0, the coordinate change on M is given by In short, a time dependent rotation of the frame {X A (t)} along o(t) produces a uniform rotation of the coordinates on each leaf Σ t .The following result shows how the rotations R A B are related to F AB .Lemma 4.9.Given the RW space of Theorem 4.5 and its flow U from Remark 4.6, we have The frames {X ′ A (t)} and {X A (t)} are related by a rotation Proof.Direct computation using (4.17) yields and so (4.21) follows.Now, consider a set of t) for all t ∈ I, and thus This is a linear system of ODE, so unique solutions exist globally given initial data R A B (t 0 ).It is a standard fact that R A B (t) are rotations for all t provided the initial data is a rotation.This follows from the fact that It only remains to define the vector fields } is an orthonormal frame at each t ∈ I.As shown above, the corresponding stereographic coordinates satisfy The first result of the lemma tells us first how to characterize F AB in terms of the flow vector U and the natural basis ∂ z A , explicitly, The second result ensures that rotating conveniently the frame along o(t) we can get rid of F AB (t) in the corresponding new stereographic coordinates.Finally, let us show how the vector field W o,X transforms under the transformation X ′ A (t) = R A B (t)X B (t) along o(t).Recall that W o,X is the vector field tangent to the curves of constant values of {λ A } and parametrized by τ .In the coordinates {τ, z A }, W o,X is given by (3.25).We already know that the vector fields ∂ z A transform as (4.24), so it is convenient to write (3.25) as Accordingly, the vector W o,X ′ associated to the frame {X ′ A (t)} along the curve o(t) reads as Therefore, using (4.20), (4.23) and (4.24) we have

Explicit form of the Killings in the new coordinates
For completeness, we also provide the expressions of the n(n + 1)/2 Killing vector fields in the chart {τ, z A }.There are several ways to approach the problem.A very direct one is to consider the following vector field anzatz where k 0A (t), k AB (t) = −k BA (t) are n(n + 1)/2 free functions of t.A direct computation shows that ξ is a Killing vector field of g if and only if the system of differential equations where we have used ε ABC 1 ...C n−2 for the n-dim antisymmetric symbol.It is immediate to check that the n(n + 1)/2 vector fields thus constructed are linearly independent, and hence a basis of Killing vectors.
Remark A.3.The usual description of constant curvature spaces as embeddings in R n+1 are recovered as follows.Observe that (A.4) and (A.5) imply κ|y| 2 + (y 0 ) 2 = 1.(A.9) Moreover dy 0 = −κdy n+1 .Thus, when κ = 0 the metric (A.6) can be viewed as the induced metric of the surface defined by (A.9) embedded in R n+1 with the metric h AB dy A dy B +κ −1 (dy 0 ) 2 (when κ = 0 we can simply take the Euclidean metric h AB dy A dy B + (dy 0 ) 2 since the graph is now simply y 0 = ±1).Note that coordinating the surface with {y A } (so that it is described as a graph in the coordinate y 0 ) does not cover the whole surface, but just a hemisphere, when κ > 0. Note also that viewing γ as an induced metric in a one-dimensional higher space requires a different treatment depending on the value of κ. ).Now, the matrix C αβ is invertible everywhere (its determinant is up to a sign that of the contravariant metric γ ♯ ).Q αβ is also invertible because of (A.3).This means that A α β is invertible at every point.Writing the inverses of C αβ , Q αβ as C αβ , Q αβ one gets as an immediate consequence of the definition of Q αβ in item (i).The inverses of C αβ and Q αβ are which is just another way of writing (A.17).The gradients of {y α } on S \ S 0 in the local coordinates {x A } are computed immediately from (A. 15)

Lemma 3 . 6 .
2) hold whenever RW.1 The spatial gradient of the umbilicity function κ vanishes on the curve o, i.e.X(κ)| o(t) = 0 for all t ∈ I o and for all X ∈ T o(t) Σ t .RW.2 The spatial Laplacian of the umbilicity function κ vanishes on o, i.e. ∆ t κ| o(t) = 0 for all t ∈ I o .From (3.14) it is immediate to check that Condition RW.1 is equivalent to C A = 0. Combining (3.14) with the trace of (3.15) at o(t) it follows that Condition RW.2 is equivalent to 2C 0 k + k = 0. Thus, we have the following lemma.Assume the setup of Proposition 3.3.The umbilicity function κ is homogeneous, i.e. there exists a function Ξ ∈ C ∞ (I o , R) such that κ = Ξ • τ = Ξ, if and only if conditions RW.1 and RW.2 hold.In such case C A = 0, C 0 = 1 2 Ξ, and the functions k and Ξ are linked by k + Ξk = 0. (3.19)

. 22 )The function ka 2 satisfies d dt ka 2 =
Perform the coordinate change λ A = az A where a(t) is any non-zero solution of the ODE ka 2 + 2ka ȧ = 0 because of (3.23) and(3.19

. 19 )
Now, if the frame {X A (t)} is transported Fermi-Walker along o, i.e.D F W o X A = 0, then the ∂ τ component of (4.19) yields Z(z A ) = 0 and then its ∂ z A component gives F AB = 0, which by the explicit expression of Z(z A ) (cf. (4.18)) also imply that F 0A (t) are constant.Conversely, if F 0A (t) are constant and F AB = 0 then Z(z A ) = 0 and thus D F W o X A = 0 for all A.

Remark 4 . 10 .+ ǫ 0 4 |x| 2 − 2 δ
If we write the RW metric in standard coordinates in the formg = −εdτ 2 + a 2 (τ ) 1 AB dx A dx B .(4.25)then the curve o(t) can be described by a set of functions o(t) := {x A = o A (t)}.If we consider the simplest coordinate transformation that sends this stereographic coordinates {x A } to stereographic coordinates {z A } centered at o(t), one can show that the metric (4.25) gets transformed into the metric of Theorem 4.5 with the following expressions for the coefficients ȯA − o A ȯB .

Remark A. 4 .
The functions {x A } of this lemma are called stereographic coordinates centered at o with frame {X A }. Given the frame {X A }, the functions {x A } are unique.Proof.The definition of Q αβ entailsQ αβ = γ grad γ y α , grad γ y β + κy α y β − y α δ β n+1 − y β δ α n+1 , (A.10)and its gradient is (in abstract index notation and letting D A be the Levi-Civita covariant derivative to γ)D A Q αβ = D A D B y α D B y β + D A D B y β D B y α + D A y α κy β − δ β n+1 + D A y β κy α − δ α n+1 =0 after inserting equation (A.1).So Q αβ is constant on S. At o we have y α = δ α 0 and γ| o grad γ y α , grad γ y β = h AB X A (y α )X B (y β ) = h AB δ α A δ β B because of (A.2).Evaluating (A.10) at o gives (A.3

Evaluating (A. 11 )
at the values µ = 0, ν = n + 1 gives (A.4) and at the values µ = ν = 0 gives h AB y A y B − 2y 0 y n+1 − κ(y n+1 ) 2 = 0, which becomes (A.5) after inserting (A.4).Evaluating (A.11) at µ = C and ν = D providesγ CD = h AB D C y A D D y B − 2D C y 0 D D y n+1 − 2D C y n+1 D D y 0 − κD C y n+1 D D y n+1⇐⇒ γ = h AC dy A dy B − dy 0 dy n+1 − dy 0 dy n+1 − κdy n+1 dy n+1 )where Hess t is the Hessian with respect to Υ t .Given that (Σ t , Υ t ) is of constant curvature k(t), Remark A.5 in Appendix A implies the existence of an open neighbourhood Σ ′ t ⊂ Σ t of o(t) where the PDE above admits a unique solution.DefineU ′ := ∪ t∈Io Σ ′ t .This is an open neighbourhood of the curve o.On this set we can define scalar functions 6into the present setting we get that the vector fields in Σ ′ t are such that {η αβ t , α < β} is a basis of the conformal Killing algebra of (Σ t , Υ t ).Their explicit form in the coordinates {λ A t } is (cf.(A.18)-(A.19)) .14)Furthermore, if we compute the Hessian of(3.13)anduse (3.2) the following equation involving only κ and k follows Hess t κ t = −(k(t)κ t + k(t))Υ t .(3.15)We are ready to prove a classification result for umbilic foliations with constant curvature leaves.Proposition 3.3 (Umbilic foliation).Let (U, I, τ, Υ) be an umbilic foliation with constant curvature leaves (cf.Definition 3.1) with umbilicity function κ and curvature k(t).Assume that U has dimension n + 1 with n ≥ 3 and let o : I o ⊂ I → U be any curve parametrized by t and {X A (t)} an orthonormal basis of T o(t) U smoothly depending on t.Then there exists an open neighbourhood V o of o, unique coordinates {τ, λ A } on V o , and unique functions C 0 after noting that gradγ x A = 1 + κ 4 |x| 2 2 h AB ∂ B , grad γ |x| 2 = 2 1 + κ 4 |x| 2 2 x B ∂ B .The result isgrad γ y 0 = −κx B ∂ B , grad γ y A = 1 + κ 4 x 2 h AB − κ 2 x A x B ∂ B , grad γ y n+1 = x B ∂ B .Inserting (A.15) and (A.20) into (A.16)yields (A.18)-(A.19)after a simple computation.It only remains to show that {ζ αβ ; α < β} is a basis.Since this set has (n + 1)(n + 2)/2 elements and this is the maximal dimension of a conformal Killing algebra in dimension n ≥ 3, it suffices to prove that they are linearly independent vector fields.Actually, it suffices to prove this on some non-empty open set.It is immediate to check that (A.18)-(A.19)are linearly independent on S \ S 0 , so the result follows.